Kjaer, 28, has been heavily linked with a move to the Premier League with Manchester United and Chelsea intrerested in securing his signature.
However, Turkish-Football claim Liverpool are now in the frame to sign the Denmark international after sending scouts to watch him.
It is suggested Reds scouts have been casting their eye over Kjaer for some time as Jurgen Klopp looks to bolster his defensive ranks.
The report adds Fenerbahce are keen to keep hold of Kjaer but will listen to offers if their £17million valuation is met.
But Kjaer recently revealed he is not interested in a move away from Turkey.